In a semiconductor junction device or a field effect device, it is often required to know the electric field and the depletion width of the device. The maximum electric field determines the maximum voltage that can be applied to the device. The depletion width determines the junction capacitance. To obtain such information analytically, one must solve the Poisson's equation.
In modern semiconductor devices, the impurity concentration is generally not uniform. Impurities are usually introduced to the semiconductor either by ion implantation or thermal diffusion to tailor the device characteristics. These impurities give rise to a nonuniform profile. Such profiles may describe a Gaussian distribution, a complementary error function, or a combination thereof. The Poisson's equations of such complicated profile are generally not easily solved by analytical methods. If numerical methods are used, the double integration, together with boundary condition determination, may require a large amount of computation time. What is needed is an efficient method to determine the electric field and capacitance of a p-n junction with a uniform impurity concentration background.